Some properties of the basic invariants of the symmetry groups $G(m,p,n)$, $B_n^m$ were described by O. I. Rudnitskii. Here we continue to study and expand these properties. We study the properties of the basis invariants of the symmetry groups of the complex polytope $\frac{1}{p}$
$\gamma_n^m$ and the generalized $n$-cube $\gamma_n^m$, as well as its subgroups $D_m^n$. We give an explicit construction of all the basis invariants of odd degree of these groups. This invariants of the symmetry groups $G(m,p,n)$, $B_n^m$ are under construction on the basis of Pogorelov's polynomials and it is possible to construct in explicit form all generators of the algebra $I^{B_n^m}$.

Conclusion:

$\Delta R_{mr}^1=J_{m(r-1)}^{*}=-2\frac{(m(p+1))!}{(mp)!}\sum_{i=1}^{n} x_i^{m(r-1)}$

Since $J_{m(r-1)}^{*}=A\sum_{i=1}^{n} x_i^{m(r-1)}$, the form $J_{m(r-1)}^{*}$ is a basic invariant of an odd degree $mr$ of group $B_n^m$.

Thus, it is proved that, on the basis of Pogorelov's polynomials, it is possible to construct in explicit form all generators of the algebra $I^{B_n^m}$.

Reference:

1. O. I. Rudnitskii, Some Properties of Basis Invariants of the Symmetry Groups $G(m,p,n)$, $B_n^m$.
Journal of Mathematical Sciences, VoL 82, No. 2, 1996

2. Anders Bjorner, Francesco Brenti, Combinatorics of Coxeter Groups, Graduate Texts in Mathematics 231,© 2005 Springer Science Business Media, Inc

3. Shephard G.C. Unitary groups generated by reflections, Can Journal Math. 1953, vol.5, page 364-383